Radiators are hot upstairs but cold downstairs

We have moved in to our new home 3 weeks ago. We have turned the heating on but only the upstairs radiators were getting hot. The downstairs radiators were ice cold.
If we turn the radiators off upstairs than downstairs is getting warm, but if we turn the upstairs radiators on again then after a few minutes the downstairs radiators getting cold again.
We have tried bleeding it first and then draining the system. But no luck.
We already had 2 servicemen from 2 different companies here to have a look on our situation, but the only thing they have done was tearing up the floorboards and the second plumber have changed the flow switch. But sitll no heating downstairs. The boiler was replaced 4 years ago so relatively it is new. The plumber said that the pump is working. The property was empty for quite a while before we have moved in 3 weeks ago.
At this point we have no idea where to go next.
Many thanks.

Balancing radiators. (1) As a first step close any rads that are getting hot and see if that heats any of the other rads. You can quickly work out which rad heats first by letting them all get cold and when they start heating up find out which pipe gets hot first, use the pipe temp to test rad for heat it is quicker than waiting for the rad panel to heat up. (2) The radiators that are nearest the pump are usually getting hot first. First make sure system is on heating only no hot water and any bypass if fitted is closed. (3) If they have a lockshield valve one end and a hand control valve the other end, make sure hand control valve is fully open and then closed just a touch. Close the lockshield valve with a small spanner on the flats of brass shaft exposed when the plastic knob is pulled off, then open it 1/8-1/4 of a turn. (4) If there is a trv then make sure they are all open and the pins are up not stuck down. (5) Then move to rad that is next one from pump and repeat, and so on until you are downstairs then open lockshield valve ¼ -1/2 a turn. By the time a valve is open by 1 turn it may as well be fully open. (6) That should work all rads. Then open bypass valve to same position it was in. If you manage to get all the rads working, once the hot water starts heating it will unbalance the rads so set the programmer to heat the hot water continuous or hot water 1.5 hrs before heating. Good luck